Finite Group Databasebeta

← #29598216

Subgroups — #29598216

Order 15616.

438 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454515C211yesmaximal, characteristic
2#21447448C212yesmaximal, complemented
3#21447429C212yesmaximal, complemented
4#21447424C211yesmaximal, characteristic, complemented
5C61×#2153C211yesmaximal, characteristic, complemented
6C2×#14519783C2211yescharacteristic
7C22×C8×D61C2211yescharacteristic
8#16840274C411yescharacteristic
9C2×#14519717C2212yes—
10#16839835—22no—
11C8×D244—22no—
12#16839789C411yescharacteristic
13#16839730C411yescharacteristic
14#16839725C411yescharacteristic
15#16839547—24no—
16#16839541—24no—
17#16839540—22no—
18C2×M32×C61C2212yes—
19C61×#405C2211yescharacteristic
20C61×#364—22no—
21C2×#11758222C2×C411yescharacteristic
22C22×C4×D61C2×C411yescharacteristic
23#14519783—22no—
24#14519783—22no—
25C2×#11758194C2×C411yescharacteristic
26C2×#11758189—21no—
27C22×#92418C2×C411yescharacteristic
28C2×C8×D61—22no—
29C2×C8×D61—21no—
30C2×C8×D61—21no—
31C2×C8×D61—21no—
32C2×C8×D61—22no—
33#14519761—21no—
34#14519757—21no—
35#14519752—21no—
36#14519747—21no—
37C4×D244—22no—
38#14519717—22no—
39#14519717—22no—
40C2×#11758185D414yescomplemented
41C4⋊#92418—22no—
42#14519689—22no—
43#14519689—22no—
44C2×M16×C61C2×C411yescharacteristic
45C22×C8×C61C2311yescharacteristic
46#117×C61C2×C411yescharacteristic
47M32×C61—22no—
48M32×C61—22no—
49C2×C16×C61—22no—
50#97×C61D412yes—
51C4×C8×C61D412yes—
52C23×D61M1611yescharacteristic
53C22×C4×C61C22×C411yescharacteristic
54C2×C22⋊D61—21no—
55C22×Dic61M1611yescharacteristic
56#11758222—22no—
57#11758222—22no—
58C2×D244—22no—
59C2×D244M1611yescharacteristic
60C2×C4×D61—21no—
61C2×C4×D61—21no—
62C2×C4×D61—22no—
63C2×C4×D61—21no—
64C2×C4×D61—22no—
65C2×Dic122M1611yescharacteristic
66M16×C61—22no—
67C2×C8×C61C22⋊C412yes—
68C2×C8×C61C2×D411yescharacteristic
69C2×C8×C61C2×D412yes—
70C2×C8×C61C2×D411yescharacteristic
71C2×C8×C61#4112yes—
72C4⋊C4×C61—21no—
73C22⋊C4×C61—21no—
74C42×C61C22⋊C412yes—
75#11758198—22no—
76#11758198—22no—
77C4⋊Dic61—22no—
78#11758194—22no—
79C2×#92418—21no—
80C2×#92418—21no—
81C2×#92418—22no—
82#11758189—42no—
83C8×D61—42no—
84C8×D61—41no—
85C8×D61—41no—
86C8×D61—41no—
87C8×D61—41no—
88C16×C61—42no—
89#11758185—24no—
90C23×C61C2×M1611yescharacteristic
91C22×D61—21no—
92C22×D61—22no—
93C22×D61—21no—
94C22×D61—42no—
95C2×C4×C61—22no—
96C2×C4×C61C2×C22⋊C411yescharacteristic
97C2×C4×C61C4×D412yes—
98C2×C4×C61C2×M1611yescharacteristic
99C2×C4×C61#13112yes—
100C22⋊D61—42no—
101C2×Dic61—21no—
102C2×Dic61—21no—
103C2×Dic61—22no—
104D244—22no—
105D244—44no—
106C4×D61—41no—
107C4×D61—41no—
108C4×D61—41no—
109C4×D61—41no—
110C4×D61—42no—
111Dic122—22no—
112C8×C61—22no—
113C8×C61#12112yes—
114C8×C61#12411yescharacteristic
115C8×C61#12711yescharacteristic
116#92418—41no—
117#92418—41no—
118#13200—611nomaximal, Sylow
119C22×C61—41no—
120C22×C61#40711yescharacteristic
121C22×C61#43512yes—
122C2×D61—22no—
123C2×D61—24no—
124C2×D61—41no—
125C2×D61—42no—
126C2×D61—41no—
127C2×D61—42no—
128C2×D61—41no—
129C4×C61—42no—
130C4×C61#39011yescharacteristic
131C4×C61#40811yescharacteristic
132C4×C61#43512yes—
133Dic61—41no—
134Dic61—41no—
135#2929—611no—
136#2153D6111yescharacteristic, complemented
137#2117—612no—
138#2112—612no—
139#2110—611no—
140C2×C61—21no—
141C2×C61—22no—
142C2×C61#195511yescharacteristic
143D61—42no—
144D61—81no—
145C2×#131—611no—
146C23×C8—611no—
147#525—611no—
148C2×M32—612no—
149C2×M32C2×D6112yes—
150#435—1222no—
151C8×D4—1222no—
152#424—611no—
153#408—611no—
154#407—611no—
155#405C2×D6111yescharacteristic
156#364—22no—
157#350—1224no—
158C22⋊C16—1224no—
159#347—1222no—
160C61#1320011yescharacteristic, complemented, Sylow
161C2×#41—611no—
162C23×C4—611no—
163#131—1222no—
164#131—1222no—
165C2×M16—611no—
166C2×M16—1221no—
167C2×M16C4×D6111yescharacteristic
168C22×C8—611no—
169C22×C8—1221no—
170C22×C8—1222no—
171C22×C8—1222no—
172C22×C8—1221no—
173C22×C8—1221no—
174C22×C8C22×D6111yescharacteristic
175#123—1221no—
176C22⋊Q8—1221no—
177#121—1221no—
178#120—1221no—
179C4×D4—1222no—
180#117C4×D6111yescharacteristic
181M32—22no—
182M32—22no—
183M32—1222no—
184M32—1222no—
185C2×C16—22no—
186C2×C16—614no—
187C4⋊C8—1222no—
188C22⋊C8—1222no—
189C22⋊C8—1222no—
190#97D24412yes—
191C4×C8C22⋊D6112yes—
192C24—611no—
193#41—1222no—
194#41—1222no—
195C2×Q8—611no—
196C2×D4—611no—
197C2×D4—1222no—
198C2×D4—1221no—
199C22×C4—611no—
200C22×C4—1221no—
201C22×C4—1221no—
202C22×C4—1222no—
203C22×C4—1221no—
204C22×C4—1222no—
205C22×C4C2×C4×D6111yescharacteristic
206M16—22no—
207M16—1222no—
208M16—2442no—
209C2×C8—1221no—
210C2×C8—1221no—
211C2×C8—1222no—
212C2×C8—2442no—
213C2×C8—2441no—
214C2×C8—2441no—
215C2×C8—2441no—
216C2×C8—2441no—
217C2×C8#1175819812yes—
218C2×C8C2×D24411yescharacteristic
219C2×C8D4×D6112yes—
220C2×C8#1175822612yes—
221C2×C8C2×C22⋊D6111yescharacteristic
222C4⋊C4—21no—
223C4⋊C4—1222no—
224C22⋊C4—21no—
225C22⋊C4—1222no—
226C22⋊C4—1222no—
227C42#1175819812yes—
228C16—42no—
229C16—1224no—
230C23—1221no—
231C23—1222no—
232C23—1221no—
233C23—2442no—
234C23M16×D6111yescharacteristic
235Q8—1222no—
236D4—1222no—
237D4—2444no—
238D4—2442no—
239C2×C4—22no—
240C2×C4—1221no—
241C2×C4—1221no—
242C2×C4—1222no—
243C2×C4—2441no—
244C2×C4—2441no—
245C2×C4—2442no—
246C2×C4—2441no—
247C2×C4—2441no—
248C2×C4C2×#1175819811yescharacteristic
249C2×C4#1451973912yes—
250C2×C4M16×D6111yescharacteristic
251C2×C4#1451978312yes—
252C8—22no—
253C8—2441no—
254C8—2441no—
255C8#1451975212yes—
256C8#1451976611yescharacteristic
257C8#1451977311yescharacteristic
258C22—41no—
259C22—1222no—
260C22—1224no—
261C22—2442no—
262C22—2441no—
263C22—2442no—
264C22—2441no—
265C22—2441no—
266C22#1683972511yescharacteristic
267C22#1683983212yes—
268C4—42no—
269C4—2441no—
270C4—2441no—
271C4#1683966311yescharacteristic
272C4#1683972811yescharacteristic
273C4#1683983212yes—
274C2—21no—
275C2—22no—
276C2—4881no—
277C2—2442no—
278C2#2144661611yescharacteristic
subgroupG/H
centerC8#14519773
commutatorC22×C4×C61C22×C4
FrattiniC22×C8C22×D61
FittingC61×#2153C2
socleC2×C61#1955
prime radical#16840274C4
cosocleC22×C8×C61C23
hypercenter#2153D61
hypocenterC61#13200
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2153

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21447448C2semidirect
3. #21447429C2semidirect
4. #21447424C2semidirect
5. C61×#2153C2semidirect
40. C2×#11758185D4semidirect
136. #2153D61semidirect
160. C61#13200semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214545152
#214474482
#214474292
#214474242
C61×#21532
#1320061